How do engineers size pumps to a given system?

How is it that engineers determine the best pumps to be used for lets say, an arrogation system? Given that you have a system with specific requirements for flow rate and pressure, how is it that one can determine the best pump for the application given just its set of non-dimensional coefficients (Cp, Cq, Ch)? That is, knowing that a pumps efficiency varies with its diameter, operating speed, flow rate, and head how can one determine the best pump that will meet the requirements while operating the most efficiently?

I see this problem as a 4 dimensional optimization problem that is going to require some type of method that requires linear algebra in order to solve it. Am I approaching this correctly or there just no systematic way to figure this out?

No matter what, you need to have tihe characteristics of your system. Then from there you take your flow requirements and back out required output pressure or head developed. That info is enough to start looking through pump curves to see if you can find a pump that will operate at its BEP. Even if you do find a pump that its BEP matches your system operating point, it still may not be the most efficient you can find.

Ideally the best way that I know of is to characterize your system. Take those numbers and go to various pump reps who can easily tell you which of their pumps would be best for your system. When you do that for multiple vendors, you can use their best pumps curves to estimate pump power required if that is your main constraint for "most efficient." There are a ton of other constraints that can come into play that will push you to one pump or another.

The amount information in terms of pump curves, etc... for multiple manufacturers would preclude any kind of analysis like you mention. It will always come down to having to look at pump curves.

The amount information in terms of pump curves, etc... for multiple manufacturers would preclude any kind of analysis like you mention. It will always come down to having to look at pump curves.

This is what I wanted to know. I have my system completely defined and know its requirements. Yet I have to choose from a variety of pumps and only have their non-dimensional coefficients, not their curves. I'm used to using manufacture's curves to match pumps to the required systems from previous internships. But now I am faced with a problem from academia where only the coefficients are provided. And now I am stuck.